Many-Body Perturbation Theory and the GW Approximation

In practical applications, we have to obtain (under certain unavoidable approximations) the self-energy operator. FYom this we calculate the QP spectrum using (5.7) and, if required, the full Green s function given by (5.5). MBPT provides a tool for such a task but, as in any other perturbation theory, we have to define the unperturbed system and the perturbation itself. In the 

The self-energy can be obtained from a self-consistent set of Dyson-like equations known as Hedin s equations  

That is, in GIT the screened Coulomb potential is calculated at the RPA level and E is just the direct product of G and W (hence the name). Also note 

In most GW applications, self-consistency is set aside, and P and S are obtained by setting G = Gks in (5.12a) and (5.12b). The interacting Green s function is then obtained by solving (5.10e) once. Furthermore, in many cases there is an almost complete overlap between the QP and the KS wavefunctions, and the full resolution of the QP equation (5.7) may be circumvented. Thus, is given as a first-order perturbation of the KS 

Inclusion of vertex corrections improves the description of the absolute position of QP energies in semiconductors [23] and the homogeneous electron gas (HEG) [24], although the amount of such corrections depends very sensitively on the model used for the vertex [25]. Vertex corrections constructed using the so-called cumulant expansion [26], reproduce the multiple-plasmon satellite structure in alkali metals [27] (the GW spectral function only shows an isolated satellite). 

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